event-study methodology: correction for cross-sectional

Event-Study Methodology: Correction for Cross-Sectional Correlation in Standardized Abnormal Return Tests* James Kolari1 Texas A&M University Seppo Pynnönen University of Vaasa, Finland

1 Correspondence: Professor James Kolari, Texas A&M University, Mays Business School, Finance Department, College Station, TX 77843-4218. Email address: [email protected] Office phone: 979-845-4803 Fax: 979-845-3884 * The main ideas in this paper were developed while Professor Pynnönen was a Visiting Faculty Fellow at the Mays Business School, Texas A&M University under a sabbatical leave funded by a senior scientist grant from the Academy of Finland. The hospitality of the Finance Department and Center for International Business Studies in the Mays Business School and the generous funding of the Academy of Finland are gratefully acknowledged.

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Event-Study Methodology: Correction for Cross-Sectional Correlation in Standardized Abnormal Return Tests Abstract

Standardized methods by Patell (1976) and Boehmer, Musumeci, and Poulsen (1991) have been

shown to outperform traditional, non-standardized tests in event studies. However, standardized

tests are valid only if there are no cross-sectional correlations between the observations’ returns.

In this paper we propose simple corrections to these test statistics to account for such correlation.

To demonstrate the usefulness of correcting for cross-sectional correlations in standardized

abnormal return tests, we conduct simulation analyses of abnormal stock return performance

using daily returns. The simulation results show that even moderate cross-sectional correlation

in the residual returns causes substantial over-rejection of the null hypothesis by the original

statistics. Results for the corrected statistics reject the null hypothesis on average at around the

nominal rate.

Key Words: Event studies; Cross-sectional correlation; Statistical simulation JEL Classification: G14; C10; C15

3

Event-Study Methodology: Correction for Cross-Sectional Correlation in Standardized Abnormal Return Tests

I. Introduction

A basic assumption in traditional event study methodology is that the abnormal returns

are cross-sectionally uncorrelated. This assumption is valid when the event day is not common

to the firms. Even in the case when the event day is common, if the firms are not from the same

industry, Brown and Warner (1982, 1985) show that use of the market model to derive the

abnormal return reduces the inter-correlations virtually to zero and, hence, can be ignored in the

analysis. Nevertheless, it is well known that, if the firms are from the same industry or have

some other commonalities, extraction of the market factor may not reduce the cross-sectional

residual correlation. Consequently, use of test statistics relying on independence understate the

standard errors and lead to severe over-rejection of the null hypothesis of no event effect when it

is true.

The traditional approach to account for correlation between returns is the so-called

portfolio method suggested by Jaffe (1974), in which the firm returns are aggregated in an

equally-weighted portfolio and the abnormal returns of the portfolio are investigated. While this

captures the contemporaneous dependency between the returns, it is generally sub-optimal.

In this regard, there have been several other attempts in the literature to solve the

contemporaneous correlation problem [See Khotari and Warner (2005) for a review]. The

Generalized Least Squares (GLS) is known to be optimal under certain assumptions, but it

requires accurate estimation of the covariance matrix of the returns, which is not always possible,

particularly if the number of firms is larger than the number of time points in the estimation

period. However, as noted above, ignoring the contemporaneous correlations may introduce

4

extensive downward bias into the standard deviation and thereby overstate the t-statistic, which

leads to over-rejection of the null hypothesis. Also, the cost of estimating the large number of

covariance parameters needed in GLS has been found to introduce even more inaccuracy into the

standard errors than it eliminates, thereby making the test results even worse [See, for example,

Malatesta (1986)]. Futhermore, Chandra and Balachandran (1990) argue that GLS is highly

sensitive to model mis-specification, which may lead to inefficient test results even if the

covariance matrix is known. They conclude that GLS should be avoided in event studies

because the correct model specification is rarely known for certain. Consequently, Chandra and

Balachandran (1990) recommend the use of nongeneralized least squares, which essentially

reduces to the portfolio tests cited above.

Particularly relevant to the present study, methods based on standardized abnormal

returns have been found to outperform those based on non-standardized returns. The most

widely used standardized methods are the Patell (1976) t-statistic and the Boehmer, Musumeci,

and Poulsen (hereafter BMP) (1991) t-statistic. However, both of these standardized tests rely on

the assumption that the abnormal returns are contemporaneously uncorrelated. At least in the

case of the Patell (1976) approach, one method of resolving the contemporaneous correlation

problem (as suggested above) is to aggregate the standardized abnormal returns using an equally-

weighted portfolio and compute the t-statistic from the portfolio returns. Unfortunately, the

portfolio method does not work in the popular BMP (1991) approach. In an attempt to resolve

potential bias in test statistics arising from cross-sectional correlations, the present paper

contributes to the event study methodology literature by deriving simple formulas that correct the

original Patell t-statistic and the original BMP t-statistic for cross-sectional correlations.

5

The remainder of this paper is organized as follows. Section II provides corrected test

statistics for abnormal returns in event studies with cross-sectional correlation between

observations. Section III discusses the simulation design. Section IV presents the empirical

results. Section V concludes.

II. Correlation Corrected Test Statistics for Standardized Abnormal Returns

Patell’s (1976) statistic is of the form

(1) 2

)4()4/()2( −

−×=−−

=m

mnAmm

nAtP ,

where A is the average of standardized abnormal returns over the sample of n firms on the event

day, and m is the number of observations (i.e., days, months, etc.) in the estimation period. The

standardized abnormal returns are calculated by dividing the event period residual by the

standard deviation of the estimation period residual, corrected by the prediction error [See, for

example, Campbell, Lo, and MacKinlay (1997, p. 160)]. Boehmer et al. (1991) estimate the

cross-sectional variance of the standardized abnormal returns and define a t-statistic (BMP t-

statistic) as

(2) s

nAtB = ,

where s is the (cross-sectional) standard deviation of the standardized abnormal returns.

If the event day is the same for the firms, the Patell and BMP t-statistics do not account for

contemporaneous return correlations. In the literature various methods have been suggested to

deal with this problem. Generalized least squares (GLS) is the optimal solution if the return

covariance matrix can be estimated accurately and the abnormal return generating model is

known [See, for example, Chandra and Balachandran (1990)]. However, these requirements are

6

rarely met. Probably the most common way to circumvent this problem is the so-called portfolio

method, where firm returns are aggregated in an equally-weighted portfolio. This method

implicitly accounts for the contemporaneous correlations. Nevertheless, in comparison to the

GLS, it does not lead to optimal estimation of the event effect. The advantage of the Patel (1976)

method as well as the BMP (1991) method is that they weight individual observations by the

inverse of the standard deviation, which implies that more volatile (i.e., more noisy) observations

get less weight in the averaging than the less volatile and hence more reliable observations. This

is roughly the idea in the GLS, where the observations are weighted by the elements of the

inverse of (residual) return covariance matrix. The BMP statistic has gained popularity over the

Patell (1976) statistic because it has been found to be more robust with respect to possible

volatility changes associated with the event. Neither of these methods, however, accounts for the

possible cross-sectional correlations that can exist when the event day is the same for the firms.

Because stock returns are typically positively correlated, ignoring such correlations leads to

underestimation of the abnormal return variance and, in turn, over-rejection of the null

hypothesis of no event effect when it is true. Consequently, we propose below simple

corrections to these statistics to account for the cross-correlations.

Implicitly, the Patell (1976) and BMP (1991) tests assume that the standardized abnormal

returns are homoscedastic and therefore have the same variance. Indeed, if there is no volatility

effect due to the event, all standardized abnormal returns would have roughly a unit variance and

lead to the Patell (1976) t-statistic. The BMP approach relaxes the no-volatility-impact, and

estimates the (common) event-day-volatility cross-sectionally with the usual sample standard

deviation. However, when the event day is the same for all firms, the standardized abnormal

returns are potentially correlated, which can bias the volatility estimates in both cases.

7

A. Single Common Event Day

Let 2Aσ be the common population variance of the standardized abnormal returns (which

equals (m-2)/(m-4) if there is no event induced variance), and let ijσ denote the population

covariance of standardized abnormal returns for securities i and j. Using simple algebra, the

variance of the mean of the standardized abnormal returns over n firms is

(3) ∑∑= ≠

+=n

i ijijAA nn 1

222 11 σσσ .

Because the variances are the same for all standardized abnormal returns, i.e., 222Aji σσσ == , the

covariances can be written as

(4) ijAijjiij ρσρσσσ 2== ,

where ijρ is the correlation of the abnormal returns of stocks i and j. As such, we can write

equation (3) as

(5) ( )ρσρσσ )1(111 2

12

22 −+=⎟⎟⎠

⎞⎜⎜⎝

⎛+= ∑∑

= ≠

nnnn

An

i ijijAA ,

where ρ is the average correlation of the abnormal returns. Note that, in order to keep the

correlation matrix positive definite, equation (5) implies that the return correlations cannot be

highly negative on average. Assuming that the event does not change the residual correlation,

the average correlation of the abnormal returns can be estimated by averaging the sample

correlations of the estimation period residuals. In the Patell (1976) statistic

)4/()2(2 −−= mmAσ , where m is the number of observations.1 Accordingly, using equation (5),

a correlation-adjusted Patell t-test, APt , becomes

8

(6) rn

trnmm

nAt PAP )1(1)1(1)4/()2( −+

=−+−−

= ,

where r is the average of the sample correlations of estimation period residuals, and Pt is the

Patell (1976) t-statistic defined in equation (1).

In the BMP t-statistic the standard deviation is estimated cross-sectionally as the square

root of the sample variance

(7) ∑=

−−

=n

ii AA

ns

1

22 )(1

1 .

Following Sefcik and Thompson (1986, p. 327), given the assumption that [ ] AiAE µ= (See

Appendix A for details), it can be easily shown that

(8) [ ] 22 )1( AsE σρ−= .

Thus, equation (7) is a biased estimator of the variance 2Aσ . Normally, because ρ is positive,

2s understates the true cross-sectional variance. Because [ ] 22 )1/( AsE σρ =− , a feasible

estimator of the variance 2Aσ is

(9) r

ssA −=

1

22 ,

where r is the average of the sample cross-correlations of the estimation period residuals.

Therefore, an estimator of the variance of the mean abnormal return A is obtained by replacing

the parameters in equation (5) by the estimators so that

(10) ))1(1(2

2 rnnss A

A −+= .

Using these results in the original BMP t-statistic given by equation (2), the correlation-adjusted

t-statistic, ABt , becomes

9

(11) rn

rtrns

nAsAt B

AAAB )1(1

1)1(1 −+

−=−+

== ,

where Bt is the BMP t-statistic given in equation (2). From equation (11) it is immediately

obvious that, if the return correlations are zero, the modified statistic reduces to the original t-

statistic.

As seen from equations (6) and (11), the severity of cross-sectional correlation in the t-

statistics is both a function of the average correlation and number of firms. It is important to

note that underestimation of the variance due to the correlation causes a more pronounced over-

rejection of the true null-hypothesis in the two-sided test than in the one-sided test. This is

obvious from Table 1, where the size problems of the unadjusted test statistics are numerically

demonstrated for one-sided and two-sided tests at the nominal 5 percent level for different

sample sizes and degrees of cross-sectional correlation.

Typically, market model residual cross-sectional correlations in intra-industry returns are

fairly low. For example, using U.S. data, Bernard (1987) finds an average correlation of 0.04 for

daily observations. However, as shown in Table 1, we find that, even with low average

correlation, problems emerge (especially in the two-sided test) at a sample size of about 10 firms

in the event study. Based on an average correlation of only 0.05, the true rejection probabilities

at the nominal 5 percent level in a sample of 10 firms for the two-sided tests are 0.10 for the

Patell test and 0.11 for the BMP test. In a larger sample of 100 firms, the true rejection

probabilities with average correlation of 0.05 are already over 0.40 in the two-sided test and

about 0.25 in the one-sided test. That is, instead of a 5 percent rejection rate, the true null

hypothesis would be rejected with more than a 40 percent (25 percent) probability in the two-

sided (one-sided) test. Thus, although the market model obviously captures a large share of the

10

return contemporaneous correlation, the remaining relatively small correlation still materially

biases the significance levels with even moderate sample sizes.

B. Clustered Common Event Days

Suppose next that we have q clusterings or groupings of the event days, where in each

group the event day is the same for the corresponding firms. Then the correlations of the non-

overlapping event-day-groups are zero and the covariance matrix of the standardized abnormal

returns is block-diagonal. Equivalently, we can think of having q industries, where all have the

same event day but the between-industry correlations are zero. In both cases the kth block

corresponds to the covariance matrix of the firms belonging to the kth group with covariance

matrix kΣ , qk ,,1K= . The average standardized abnormal return is ∑ == q

k kk Ann

A1

1 , where

kA is the average standardized abnormal return in subgroup k, and kn is the number of firms in

subgroup k. Again assuming that within each sub-group the variances of the standardized returns

are the same, the variance of kA is of the form shown in equation (3). Consequently, the

variance of the average abnormal return over all firms becomes

(12) ∑ ∑= =

−+==q

k

q

kkkkkAkA nn

nn

n k1 1

22

222

2 ))1(1(11 ρσσσ ,

where 2kAσ is the variance of the average abnormal returns in group k, 2

kσ is the variance of the

standardized abnormal return in group k, and kρ is the average abnormal return correlation in

group k.2

For the sake of simplicity, assume that the estimation periods for all firms are the same.

Then in the Patell (1976) statistic )4/()2(2 −−= mmkσ for all qk ,,1K= . The average of the

cross-sectional sample correlations of the residuals, kr , for group k are defined as

11

(13) ∑∑=

≠=−

=k kn

i

n

ijj

kijkk

k rnn

r1 1

,)1(1 ,

where kijr , is the sample correlation of the market model residuals of returns i and j in group k

calculated over the sample period. Replacing kρ in equation (12) by the estimator (13) gives the

correlation adjusted Patell t-statistic

(14) ( ) ( )∑∑ ==

−+=

−+

−−×= q

k kkk

Pq

k kkk

APrnn

ntrnn

mmnAt

11)1(1)1(1

)2/()4(,

where Pt is the Patell (1976) t-statistic. We can write further

(15) ( )

( ),~)1(1

)1()1(111

rnn

rnnnrnnq

kkkk

q

kkkk

−+=

−+=−+ ∑∑==

where

(16) ∑=

−−

=q

kkkk rnn

nnr

1)1(

)1(1~ ,

such that r~ is the average sample correlations over the whole (block) correlation matrix with

between-block sample correlations set to zero (i.e., a restricted average correlation estimator).

Using these notations, we can write equation (14) in the same form as equation (6), such that

(17) rn

t

rnmmnAt P

AP ~)1(1~)1(1)4/()2( −+=

−+−−= ,

where the only difference is that the unrestricted average correlation estimator, r , is replaced by

the restricted average correlation estimator, r~ , defined in equation (16).

12

Estimating the abnormal return variance with cross-sectional estimator (7) as in the BMP

t-statistic, and using similar methods provided in Appendix A, the expected value of the

estimator in the grouped data can be straightforwardly shown to be

(18) [ ] ( )

( ),~1

)1(1111

1

2

1

22

ρσ

ρσ

−=

⎟⎠⎞

⎜⎝⎛ −+−

−= ∑

=

A

q

kkkkA n

nn

nsE

where ∑ =−

−= q

q kkk nnnn 1

)1()1(

1~ ρρ is the average correlation over the whole correlation

matrix including the zero correlations. Consequently, as in formula (9), a feasible estimator of

2Aσ is

(19) r

ssA ~1

22

−=

where r~ is the restricted average sample correlation defined in equation (16). The

corresponding correlation adjusted t-statistic for the grouped data is of the form (11) with r

replaced again by the restricted average correlation estimator r~ . That is, we have

(20) rn

rtt BAB ~)1(1

~1−+

−= .

III. Simulation design

A. Samples

We follow the design setup by Brown and Warner (1985) by constructing 250

independently drawn portfolios of sizes n = 50, 30, or 10 securities each. Because we are

interested in the effect of cross-sectional correlation on the test statistic, we restrict the analyses

to one industry. We selected the two-digit SIC industry code 36, which is one of the largest in

13

terms of the number of firms with 1058 securities available on CRSP. According to the U.S.

Department of Labor (www.osha.gov), this industry consists of firms in “Electronic and Other

Electrical Equipment and Components, except Computer Equipment.” The total sample period

covers CRSP daily returns from January 3, 1990 to December 31, 2004. In each round of

simulation, initially a common randomly drawn event day is selected, which is set as date “0,”

and then a sample of n securities are selected without replacement. In order for a security to be

included in the sample, it must have at least 50 returns in the common estimation period (-249

through -11) and no missing returns in the 30 days surrounding the event date (-19 to +10).

B. Abnormal Returns

First, we generate fixed abnormal returns on day 0 by adding to the day 0 residual return

an abnormal return of 0%, 0.5%, 1%, 2%, and 3%. Second, we allow for variance changes by

adding to the day 0 residual return abnormal returns generated from the multivariate normal

distribution with constant mean vectors of 0%, 0.5%, 1%, 2%, and 3% and a 5050× covariance

matrix Σ equal to the estimation period cross-sectional covariance matrix of the market model

residuals. We increase the variance (covariances) by factor c in the manner described in

Boehmer et al. (1991), such that the event induces additional variance-covariance is Σc , where c

is a constant equal to 0, 0.5, 1, or 2. Thus, the total covariance matrix in the event day 0 is

Σ+=Σ )1( cc , which implies that c = 0 corresponds to a no event-induced variance and c = 2 a

variance of 3 times the non-event variance, or 73.13 ≈ times the non-event standard deviation.

It is notable that the correlations of the residual returns do not change with these variance-

covariance increments. In the estimation of the market model, we use the equally-weighted

version of the SP500 index as the market portfolio.

14

C. Test Statistics

In addition to the unadjusted Patell (1976) and BMP (1991) statistics given in equations

(1) and (2), respectively, and their adjusted extensions given in equations (6) and (11),

respectively, we report results for the traditional cross-sectional t-statistic

(21)

∑=

=n

ii

cs

sn

nAt

1

21,

where 2is is the market model residual variance of security i. Additionally, we report the results

for the standard portfolio method t-statistic

(22) sAt pf = ,

where

(23) ∑−

−=

−−=11

249

2, )ˆˆ(

2371

ttmt RRs βα ,

with ∑ == tn

i tit

t Rn

R1 ,

1 corresponding to the equally-weighted portfolio return of tn securities,

tmR , is the equally-weighted SP500 return, and α̂ and β̂ are the OLS estimates of the market

model parameters. Recall that this portfolio approach implicitly accounts for the cross-sectional

correlations.

IV. Results

The simulation results are reported in Tables 2 through 6. Table 2 reports sample

statistics under the null hypothesis of no event effect. The overall average of the return cross-

correlations in the simulations is 0.077 for the samples of 50 securities and 0.078 for the samples

15

of 30 and 10 securities. The average residual cross-correlation after extracting the market factor

is 0.033 for the 50 securities samples and 0.036 for the 30 and 10 securities samples. All the

average t-statistics are close to zero as expected due to no event effect being imposed. Although

the average residual correlation is fairly low, its effect is quite dramatic with respect to the

distributional properties of the unadjusted t-statistics. The third column shows that for all sample

sizes (50, 30, or 10) the standard deviations of the traditional, Patell, and BMP statistics are

typically more than 1.5 times the theoretical value of one under the null hypothesis. On the other

hand, for the portfolio, adjusted Patell, and adjusted BMP methods the average standard

deviations are close to the theoretical value of one, except for the adjusted BMP method in the

case of n = 10 (small sample size) security portfolios. Thus, these preliminary results clearly

demonstrate the fact that ignoring even small (average) correlation may substantially bias the

distributional properties of the test statistics via underestimation of the true (residual) return

variability.

Table 3 reports rejection rates using samples of 50, 30, and 10 securities at the 5 percent

level for one- and two-tailed tests for the test statistics when the event has no mean effect but

may increase variability. The second and third columns report results where there is neither a

mean effect nor a volatility increase due to the event. The results clearly indicate that for all

sample sizes, due to the correlation, the traditional t-statistic, Patell t-statistic, and BMP statistic

all over-reject the null hypothesis with rejection rates normally 2 to 3 times the nominal rate of

0.05 in the one-tailed test and 2 to 5 times in the two-tailed test. As equations (6) and (11)

predict, the over-rejection rates are more pronounced in larger samples, which is also supported

by the empirical results of Table 3, where for n = 50 the rejection rates for the unadjusted

statistics are from 0.105 to 0.156 in the one-tailed test and from 0.208 to 0.248 in the two-tailed

16

test. Contrasting these with respect to the theoretical rejection rates similar to Table 1, the

theoretical value for the (unadjusted) Patell statistic in the one-tailed test is

( ) 163.0)033.0491/96.11 ≈×+Φ− and in two-tailed test

( ) 242.0033.0491/645.11(2 ≈×+Φ−× , where )(⋅Φ is the standard normal distribution

function. These theoretical rejection rates compare closely with the simulated empirical

rejection rates for the one-tailed and two-tailed tests, which are 0.158 and 0.248, respectively.

For the (unadjusted) BMP statistics the corresponding theoretical rejection rates are 0.168 and

0.251, respectively, while the empirical estimates are 0.128 and 0.244. Here we see that the one-

tailed theoretical value is to some extent under-estimated.

The remaining columns in Table 3 report the results with event-induced variability. In

sum, the overall finding is that the over-rejection gets worse as the variability increases. For

example, the Patell statistic rejects the null hypothesis (i.e., no event-induced mean effect) for 50

securities and event-induced variability factor c = 2.0 with probability 0.520, which makes the

test useless.

The last three methods (i.e., portfolio, adjusted Patell, and adjusted BMP) in Table 3 are

supposed to account for the cross-sectional correlations. In the case of n = 50 securities and no

event-induced additional variability, the rejection rates are from 0.040 through 0.064, and hence

closely approximate the nominal rate of 0.05. In the smaller samples the estimates are less

accurate. For example, with n = 10 securities the rejection rates for the adjusted Patell and

adjusted BMP are about 0.10 for the two-sided tests. In sum, except for very small samples,

these results indicate that the proposed simple corrections in these statistics remove the bias in

the rejection rates.

17

Table 3 also reports the results when there is event-induced variability but no mean effect.

Because the statistics not accounting for cross-sectional correlation (i.e., traditional, unadjusted

Patell, and unadjusted BMP) already produce over-rejection even for no event-induced

variability, we focus here on the statistics that account for correlation (i.e., portfolio method,

adjusted Patell, and adjusted BMP). The last three lines of each of the sample size panels in

Table 3 clearly demonstrate that, while the portfolio method and adjusted Patell method account

for the correlation effect, they do not capture the event-induced variability. In all cases over-

rejection increases as a function of the increased variability. Even for c = 0.5 the rejection rate is

usually 2 to 3 times the nominal rate. The adjusted BMP statistic is the only one for which in the

presence of event-induced variability the estimated rejection rates are reasonably close to the

nominal rate of 0.05. If the number of firms in the event study is small, then the rejection rate

becomes inaccurate, which obviously is due to the inaccuracy in the variance estimation from a

sample of 10 (correlated) observations. Overall, the results in Table 3 indicate that methods not

accounting for correlation tend to heavily over-reject the null hypothesis when it is true, and

methods accounting for the correlation but not the event-induced variability are vulnerable to the

variability increase, thereby resulting in considerable over-rejection of the null hypothesis of no

mean effect due to the event. For all but very small sample sizes, the adjusted BMP statistic

appears to be the only one that captures both the correlation and the event-induced variability.

Tables 4 through 6 report rejection rates (power) when there are both an event-induced

mean effect and a variability effect for the methods accounting for cross-correlation (i.e.,

portfolio method, adjusted Patell, and adjusted BMP). The power results are not relevant for the

traditional and unadjusted Patel or BMP because of the over-rejection of the null hypothesis of

no mean event effect in the presence of correlation. For the same reason (i.e., over-rejection) the

18

power comparisons are also not relevant for the portfolio method and adjusted Patel in the case

of event-induced variability. Consequently, we have not reported them in the tables.

The second column of the Tables 4 through 6 shows the results with no event-induced

variance. The adjusted Patell and adjusted BMP methods detect the false null hypothesis at

about the same rate. The advantage of the latter is that it is more robust towards event-induced

volatility. The simulation results in the second columns (c = 0.0) of Tables 4 through 6 also

confirm the well- known fact that the portfolio method is less powerful than the other two

methods. Depending on the value of the abnormal return and number of securities, the reduction

of power of the portfolio method is from 10 to 60 percent (normally around 30%) compared to

the other two methods. The adjusted Patell and adjusted BMP methods are about equally

powerful even in small samples, although one would expect the adjusted Patell method to have

more power because of the presumably more accurate return variance estimation. The BMP

statistic is the only one which accounts for event-induced variance. As the variability increases,

the observations become more noisy, which decreases the accuracy of inference. Also, there is a

decrease in the power of the BMP test for more volatile abnormal returns.

V. Conclusions

In this paper we have demonstrated via simulation that, using the traditional standardized

return test statistics, even moderate cross-sectional correlation in an event study causes

substantial over-rejection of the null hypothesis of no event mean effect. We have proposed

simple corrections to the popular Patell (1976) and Boehmer, Musumeci, and Poulsen (BMP)

(1991) statistics to account for the correlation. Our simulations show that, when there is no

event-induced volatility increase, both of these corrected test statistics are approximately equally

19

powerful and reject the null hypothesis at the correct nominal rate when the null hypothesis is

true. However, the Patell statistic is sensitive to event-induced volatility and rejects the null

hypothesis too often. The adjusted BMP statistic is robust against the event-induced volatility.

However, in order to get reliable results with the BMP method, there must be enough firms for

the cross-sectional volatility estimation.

20

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Handbooks of Corporate Finance: Empirical Corporate Finance, Chapter 1 (Elsevier/North-

Holland, Amsterdam).

Malatesta, P. H., 1986, Measuring abnormal performance: The event parameter approach using

joint generalized least squares, Journal of Financial and Quantitative Analysis 21, 27–38.

Patell, J., 1976, Corporate forecasts of earnings per share and stock price behavior: Empirical

tests, Journal of Accounting Research 14, 246–276.

21

Sefcik, S. E., and R. Thompson, 1986, An approach to statistical inference in cross-sectional

models with security abnormal returns as dependent variable, Journal of Accounting

Research 24, 316–334.

22

Footnotes

1. If the estimation periods are different for each firm, then )4/()2(2 −−= jjj mmσ , where jm

is the number of observations on the estimation period of the jth firm. In this case equation (5)

holds only approximately. Nevertheless, if the estimation periods are reasonably long, such that

22 )4/()2()4/()2( jjjiii mmmm σσ =−−≈−−= , then substituting 2iσ and 2

jσ by

∑ =−−= n

j jjA mmn 1

2 )4/()2(1σ [see Patell (1976)] in formula (4) introduces only negligible bias

in the standard error of the mean abnormal return.

2. In the more general case where jkm is the length of the estimation period of firm j in group k,

knj ,,1K= , qk ,,1K= , the variances are )4/()2()(2 −−= jkjkj mmkσ . As discussed in

footnote 1, if the estimation periods have reasonably many observations,

)()4/()2()4/()2()( 22 kmmmmk jjkjkikiki σσ =−−≈−−= , approximating individual variances

)(2 kiσ with the average ∑ =−−= kn

j jkjkk

k mmn 1

2 )4/()2(1σ , which again introduces only

negligible bias into the standard error of the average abnormal return given in equation (12).

23

Appendix

Here we derive the expected value of cross-sectional variance estimator with non-zero

cross-correlations. Suppose that we have n cross-sectional abnormal returns nAA ,,1 K that are

identically distributed with [ ] AiAE µ= and ( )[ ] 22 )var( AiAi AAE σµ ==− the same for all

ni ,,1 K= , and ijji AA σ=),cov( (i.e., not independent). Then we can write ijAij ρσσ 2= , where

ijρ is the correlation between the abnormal returns i and j. The standard estimator for 2Aσ is

(A1) ∑=

−−

=n

ii AA

ns

1

22 )(1

1 ,

where ∑=

=n

iiA

nA

1

1 is the mean abnormal return. Then the expected value of 2s , [ ]2sE , is

[ ] 22 )1( AsE σρ−= , where ∑∑= ≠−

=n

i ijijnn 1)1(

1 ρρ is the average correlation between the returns.

This can be easily seen as follows (c.f. Sefcik and Thompson, 1986). Define

(A2) [ ] ∑=

−−

=n

ii AAE

nsE

1

22 )(1

1

and

(A3) ( )[ ]

.)())((2)(

)()()(22

22

AAAiAi

AAii

AEAAEAEAAEAAE

µµµµµµ

−+−−−−=

−−−=−

The covariance term in the middle is

24

(A4)

.1

11

))((1))((

1

2

1 1

2

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+=

==

−−=−−

∑

∑ ∑

∑

≠=

= =

=

n

ijj

ijA

n

j

n

jijAij

n

jAjAiAAi

n

nn

AAEn

AAE

ρσ

ρσσ

µµµµ

The last term on the right-hand-side of equation (A3) becomes

(A5)

( ).)1(11

)1(1)1(1

))((1)(1

)(1)(

2

1 12

2

1 112

22

2

1

2

ρσ

σσ

µµµ

µµ

−+=

∑ ∑−

−+=

∑ ∑ −−∑ +−=

⎟⎠

⎞⎜⎝

⎛ −∑=−

=≠=

=≠==

=

nn

nnnnn

n

AAEn

AEn

An

EAE

A

n

j

n

jkk

ijA

n

j

n

jkk

AkAj

n

jAj

A

n

jjA

Using equations (A2)–(A5) in equation (A1), we finally get

(A6) ( ) ( )( )

).1(

)1()1(1

1

)1(1)1(12(1

1

))1(1(1121

1

)(1

1)(

2

2

222

1

2

1

22

1

22

ρσ

ρσ

ρσρσσ

ρσρσσ

−=

−−−

=

−++−+−−

=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−++

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+−

−=

−−

=

∑ ∑

∑

=≠=

=

A

A

AAA

n

iA

n

ijj

ijA

A

n

ii

nn

nnnn

nnnn

AAEn

sE

Table 1

True rejection probabilities at the nominal 5 percent level for one-sided and two-sided unadjusted Patell and Boehmer, Musumeci, and Poulsen (BMP) t-tests of average abnormal returns when the returns are cross-sectionally correlated.

Number

of Average correlation ( ρ ) Average correlation ( ρ ) firms (n) 0.00 0.01 0.05 0.10 0.15 0.20 0.00 0.01 0.05 0.10 0.15 0.20

Panel A. Patell t-statistic (one-tailed) Panel B. Patell t-statistic (two-tailed) 5 0.05 0.05 0.07 0.08 0.10 0.11 0.05 0.05 0.07 0.10 0.12 0.14

10 0.05 0.06 0.09 0.12 0.14 0.16 0.05 0.06 0.10 0.16 0.20 0.2420 0.05 0.07 0.12 0.17 0.20 0.23 0.05 0.07 0.16 0.25 0.32 0.3730 0.05 0.07 0.15 0.20 0.24 0.26 0.05 0.08 0.21 0.32 0.40 0.4550 0.05 0.09 0.19 0.25 0.28 0.31 0.05 0.11 0.29 0.42 0.50 0.55100 0.05 0.12 0.25 0.31 0.34 0.36 0.05 0.16 0.42 0.55 0.62 0.67200 0.05 0.17 0.31 0.36 0.38 0.40 0.05 0.26 0.55 0.67 0.72 0.76

Panel C. BMP t-statistic (one-tailed) Panel D. BMP t-statistic (two-tailed) 5 0.05 0.05 0.07 0.09 0.12 0.14 0.05 0.06 0.08 0.12 0.15 0.19

10 0.05 0.06 0.09 0.13 0.16 0.19 0.05 0.06 0.11 0.18 0.24 0.2920 0.05 0.07 0.13 0.18 0.22 0.25 0.05 0.07 0.17 0.27 0.36 0.4230 0.05 0.07 0.15 0.21 0.26 0.29 0.05 0.09 0.22 0.35 0.43 0.5050 0.05 0.09 0.19 0.26 0.30 0.33 0.05 0.11 0.30 0.44 0.53 0.59100 0.05 0.12 0.26 0.32 0.35 0.37 0.05 0.17 0.43 0.57 0.65 0.70200 0.05 0.17 0.31 0.37 0.39 0.41 0.05 0.26 0.56 0.68 0.74 0.78

Table 2

Sample statistics for event tests from 250 simulated portfolios of n = 50, 30, and 10 securities under no event effects when the residual returns are correlated.

n = 50 Mean Std. Min Max Traditional t-test [Eq. (21)] -0.063 1.449 -4.646 4.964 Patell test [Eq. (1)] 0.059 1.574 -4.562 6.072 BMP test [Eq. (2)] -0.019 1.550 -4.722 5.221 Portfolio method [Eq. (22)] 0.003 1.081 -3.528 7.233 Adjusted Patell [Eq. (6)] 0.027 1.022 -3.256 4.213 Adjusted BMP [Eq. (11)] -0.020 0.977 -3.810 2.747 Average return cross-correlation 0.077 0.050 0.015 0.204 Average residual cross-correlation 0.033 0.025 0.006 0.127 n = 30 Mean Std. Min Max Traditional t-test [Eq. (21)] -0.144 1.523 -6.057 4.959 Patell test [Eq. (1)] 0.031 1.531 -4.560 6.211 BMP test [Eq. (2)] -0.061 1.512 -5.430 4.897 Portfolio method [Eq. (22)] -0.019 1.068 -2.926 4.251 Adjusted Patell [Eq. (6)] 0.025 1.097 -2.774 3.412 Adjusted BMP [Eq. (11)] -0.041 1.071 -3.470 3.245 Average return cross-correlation 0.078 0.056 0.013 0.274 Average residual cross-correlation 0.036 0.031 0.003 0.179 n = 10 Mean Std. Min Max Traditional t-test [Eq. (21)] -0.173 1.442 -6.309 2.126 Patell test [Eq. (1)] -0.047 1.315 -3.906 3.422 BMP test [Eq. (2)] -0.139 1.648 -9.034 2.591 Portfolio method [Eq. (22)] 0.004 1.064 -3.222 2.560 Adjusted Patell [Eq. (6)] -0.053 1.139 -3.656 2.247 Adjusted BMP [Eq. (11)] -0.113 1.347 -6.478 2.518 Average return cross-correlation 0.078 0.061 -0.003 0.261 Average residual cross-correlation 0.036 0.037 -0.008 0.182

27

Table 3

Average rejection rates at the 5% significance level of the null hypothesis of no mean event effect in the presence of factor 0, 0.5, 1.5, and 2.0 increases in event-induced variance-

covariance for 250 random portfolios of n = 50, 30 and 10 securities.a

Event-induced variance-covariance factor c, Σ+=Σ )1( cc

c = 0.0 c = 0.5 c = 1.0 c = 2.0 Test statistic (n = 50) 1-tail 2-tail 1-tail 2-tail 1-tail 2-tail 1-tail 2-tail Traditional t-test [Eq. (21)] 0.108 0.208 0.116 0.200 0.124 0.204 0.124 0.212Patell test [Eq. (1)] 0.156 0.248 0.208 0.372 0.256 0.424 0.296 0.520BMP test [Eq. (2)] 0.128 0.244 0.152 0.252 0.152 0.240 0.148 0.228Portfolio method [Eq. (22)] 0.040 0.052 0.112 0.104 0.156 0.172 0.216 0.304Adjusted Patell [Eq. (6)] 0.052 0.064 0.080 0.128 0.120 0.208 0.164 0.280Adjusted BMP [Eq. (11)] 0.044 0.056 0.044 0.052 0.052 0.048 0.048 0.056 c = 0.0 c = 0.5 c = 1.0 c = 2.0 Test statistic (n = 30) 1-tail 2-tail 1-tail 2-tail 1-tail 2-tail 1-tail 2-tail Traditional t-test [Eq. (21)] 0.084 0.188 0.108 0.164 0.112 0.148 0.128 0.152Patell test [Eq. (1)] 0.104 0.144 0.212 0.336 0.244 0.412 0.284 0.504BMP test [Eq. (2)] 0.112 0.168 0.128 0.184 0.136 0.176 0.136 0.204Portfolio method [Eq. (22)] 0.056 0.060 0.120 0.148 0.168 0.236 0.208 0.316Adjusted Patell [Eq. (6)] 0.064 0.080 0.136 0.144 0.172 0.252 0.224 0.368Adjusted BMP [Eq. (11)] 0.052 0.060 0.048 0.064 0.060 0.068 0.060 0.072 c = 0.0 c = 0.5 c = 1.0 c = 2.0 Test statistic (n = 10) 1-tail 2-tail 1-tail 2-tail 1-tail 2-tail 1-tail 2-tail Traditional t-test [Eq. (21)] 0.084 0.096 0.092 0.120 0.076 0.112 0.132 0.116Patell test [Eq. (1)] 0.108 0.128 0.164 0.192 0.216 0.300 0.248 0.364BMP test [Eq. (2)] 0.124 0.180 0.096 0.128 0.100 0.124 0.124 0.136Portfolio method [Eq. (22)] 0.056 0.080 0.124 0.096 0.148 0.172 0.176 0.288Adjusted Patell [Eq. (6)] 0.060 0.108 0.092 0.140 0.144 0.208 0.192 0.296Adjusted BMP [Eq. (11)] 0.080 0.092 0.060 0.088 0.060 0.084 0.072 0.084

a As discussed in the text, the variance (covariances) are increased by factor c along lines described in Boehmer et al. (1991), such that the event induces additional variance-covariance is Σc , where c is a constant equal to 0, 0.5, 1, or 2 (i.e., the total covariance matrix in the event

day 0 is Σ+=Σ )1( cc , which implies that c = 0 corresponds to a no event-induced variance and c

= 2 a variance of 3 times the non-event variance, or 73.13 ≈ times the non-event standard deviation).

28

Table 4

Average rejection rates for selected test statistics for 250 randomly selected portfolios of n = 50 securities in a one-tailed test at the 5% significance level with 0.5%, 1%, 2%, and 3% abnormal

returns and a factor of 0, 0.5, 1.5, and 2.0 increase in event-induced variance-covariance.a

Rejection rates for one-tailed tests at 5% level

Event-induced variance-covariance factor c,

Σ+=Σ )1( cc Test statistic c = 0.0 c = 0.5 c = 1.0 c = 2.0 Panel A. Abnormal return 0.5% Portfolio method [Eq. (22)] 0.112 n.a n.a n.a Adjusted Patell [Eq. (6)] 0.168 n.a n.a n.a Adjusted BMP [Eq. (11)] 0.132 0.112 0.116 0.096 Panel B. Abnormal return 1.0% Portfolio method [Eq. (22)] 0.236 n.a n.a n.a Adjusted Patell [Eq. (6)] 0.404 n.a n.a n.a Adjusted BMP [Eq. (11)] 0.356 0.236 0.192 0.152 Panel C. Abnormal return 2.0% Portfolio method [Eq. (22)] 0.572 n.a n.a n.a Adjusted Patell [Eq. (6)] 0.712 n.a n.a n.a Adjusted BMP [Eq. (11)] 0.756 0.548 0.420 0.312 Panel D. Abnormal return 3.0% Portfolio method [Eq. (22)] 0.812 n.a n.a n.a Adjusted Patell [Eq. (6)] 0.896 n.a n.a n.a Adjusted BMP [Eq. (11)] 0.888 0.800 0.704 0.528



= 2 a variance of 3 times the non-event variance, or 73.13 ≈ times the non-event standard deviation). Note that n.a = not applicable, as the portfolio method and the Patell method do not account for event-induced variance-covariance.

29

Table 5

Average rejection rates for selected test statistics for 250 randomly selected portfolios of n = 30 securities in one-tailed test at the 5% significance level with 0.5%, 1%, 2%, and 3% abnormal

returns and a factor of 0, 0.5, 1.5, and 2.0 increase in event-induced variance-covariance.a Rejection rates for one-tailed tests at 5% level


Σ+=Σ )1( cc Test statistic c = 0.0 C = 0.5 c = 1.0 c = 2.0 Panel A. Abnormal return 0.5% Portfolio method [Eq. (22)] 0.096 n.a n.a n.a Adjusted Patell [Eq. (6)] 0.160 n.a n.a n.a Adjusted BMP [Eq. (11)] 0.152 0.132 0.120 0.108 Panel B. Abnormal return 1.0% Portfolio method [Eq. (22)] 0.172 n.a n.a n.a Adjusted Patell [Eq. (6)] 0.316 n.a n.a n.a Adjusted BMP [Eq. (11)] 0.284 0.236 0.204 0.156 Panel C. Abnormal return 2.0% Portfolio method [Eq. (22)] 0.412 n.a n.a n.a Adjusted Patell [Eq. (6)] 0.620 n.a n.a n.a Adjusted BMP [Eq. (11)] 0.640 0.488 0.376 0.328 Panel D. Abnormal return 3.0% Portfolio method [Eq. (22)] 0.708 n.a n.a n.a Adjusted Patell [Eq. (6)] 0.864 n.a n.a n.a Adjusted BMP [Eq. (11)] 0.860 0.724 0.624 0.504




30

Table 6

Average rejection rates for selected test statistics for 250 randomly selected portfolios of n = 10 securities in one-tailed test at the 5% significance level with 0.5%, 1%, 2%, and 3% abnormal

returns and a factor of 0, 0.5, 1.5, and 2.0 increase in event-induced variance-covariance. Rejection rates for one-tailed tests at 5% level


Σ+=Σ )1( cc Test statistic c = 0.0 c = 0.5 c = 1.0 c = 2.0 Panel A. Abnormal return 0.5% Portfolio method [Eq. (22)] 0.100 n.a n.a n.a Adjusted Patell [Eq. (6)] 0.104 n.a n.a n.a Adjusted BMP [Eq. (11)] 0.152 0.100 0.100 0.100 Panel B. Abnormal return 1.0% Portfolio method [Eq. (22)] 0.172 n.a n.a n.a Adjusted Patell [Eq. (6)] 0.220 n.a n.a n.a Adjusted BMP [Eq. (11)] 0.200 0.180 0.156 0.124 Panel C. Abnormal return 2.0% Portfolio method [Eq. (22)] 0.304 n.a n.a n.a Adjusted Patell [Eq. (6)] 0.460 n.a n.a n.a Adjusted BMP [Eq. (11)] 0.420 0.296 0.272 0.220 Panel D. Abnormal return 3.0% Portfolio method [Eq. (22)] 0.444 n.a n.a n.a Adjusted Patell [Eq. (6)] 0.612 n.a n.a n.a Adjusted BMP [Eq. (11)] 0.684 0.560 0.432 0.404




event-study methodology: correction for cross-sectional

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